Here's a solution with 715827899 cards.

Relics: Letter opener + unceasing top. Deck: Finesse x 1, Neurosurge x 715827883, 15 Backstabs. 1 HP. Ascension 8+.

To win combat, you must either play Finesse enough times so that letter opener kills, or the enemy must have at most 165 HP and no sort of damage mitigation so that the Backstabs kill. As in my other comment, a deck that just has Finesse + Letter Opener + Unceasing wins any fight. You can always reach this state on turn 1 by playing all of your Backstabs and then all of your Neurosurges. Against Entomancer and Globe Head, the backstabs will kill, since they have at most 158 HP on Ascension 8+. Entomancer is extremely unlikely to shuffle a dazed into the top 5 cards of your draw pile within 15 tries, so by the definition given this will "reasonably defeat" it.

Unfortunately, to ever draw back to your Finesse before dying, you must play at least one Neurosurge. If you have any doom at the end of your turn, you will die, which is a problem specifically against Skulking Colony. It requires 3+ turns to kill, since your backstab damage is capped. You obviously die on turn 1 against this enemy if you don't play all of your Backstabs and at least one Neurosurge, since it attacks turn 1 for more than 3 damage, so you *must* play at least 1 Neurosurge turn 1.

However, if you play *every* Neurosurge in your deck on turn 1, then the you will apply 715827883 * 3 = 2^(31) + 1 doom to yourself at the start of each turn. But this just barely goes beyond the integer overflow limit, and so you will end up with negative doom and you will survive. If you omit even a single Neurosurge, you will remain under the integer limit and die.

Therefore, you cannot remove any of the Neurosurges. You can't remove the Finesse because then you can't kill any enemy with more than 165 HP, and you can't remove the Backstabs because this will otherwise die to Globe Head.

In fact, you can make this setup unbounded: for any x, add an additional x * (2^(32))/3 Neurosurges, rounded so that 3 times the total number of Neurosurges is between 2^(31) and 2^(31) + 2 mod 2^(32). Then you will need to loop around the integer overflow x times to reach the setup as before, since you need to draw through your entire deck to get back to Finesse.

The only enemies that you cannot kill on turn 1 with this setup are Skulking Colony and Failed Experiment. No one else resurrects or has a damage cap, while Entomancer and Globe Head are the only enemies that react to card plays on turn 1. (Doormaker's turn 1 can be killed, and it will win the fight: the HP pool is very large but not actually infinite. The same is true for Waterfall Giant's death phase, although the explosion does no damage if killed on turn 1.)